Methods and arrangements for detecting weak signals

ABSTRACT

The invention provides a method and an arrangement for detecting moving point-targets within a large set of noisy measurements. The method is based on Bayesian model selection where the measurements containing targets are modeled with their physical trajectories and the non-target measurements are modeled with the statistical distribution of measurements containing no targets. An a posteriori probability density function is utilized together with a optimization algorithm specifically designed for this problem. Advantages of the invention involve a numerically efficient formulation of the a posteriori probability density, combined with the optimization algorithm. The main applications of the invention are in detecting moving targets within e.g., radar, sonar, lidar and telescopic measurements. The method is also applicable for multi-instrument data fusion.

TECHNICAL FIELD

The invention concerns generally the detection of a desired featuresfrom among a plurality of measurements, only a small part of which arerelated to the desired features. In particular the invention concernsthe use of a search algorithm and an electronic apparatus to separatemeaningful measurements from noise and other undesired measurements.

BACKGROUND OF THE INVENTION

Very few receivers of electromagnetic signals can operate in an idealway, in the sense that the output of the signal reception stage wouldonly consist of meaningful parts of the actual signal to be received. Inpractically all cases the receiver will simultaneously receive alsounwanted signals, such as simultaneous transmissions from others thanthe source of the desired signal, as well as random noise. Alsocomponents of the receiver itself generate noise, which is summed to theactual signals at the output of the reception stage. The problem ofseparating the desired signal from noise has certain universal, commonlyapplicable features regardless of what purpose (e.g. communications,remote sensing, etc.) the signal serves.

The traditional approach to separating the desired signal from noise atthe reception stage is based on filtering. For example, if a carrierfrequency of the desired signal is known, the receiver may use a bandpass filter to reject signals at frequencies that differ from the knowncarrier frequency more than half the width of a relatively narrow passband. Filtering in the time space means only qualifying receiver outputthat occurs within a time interval that is known (or assumed) tocorrespond to the desired signal. Matched filters are devices thatcorrelate the received signal with some code that is known to occur inthe desired transmission, and so on. However, some noise will alwayshave characteristics similar to those of the desired signal with certainaccuracy, and hence despite all filtering, the output of the receptionstage will always contain also unwanted signal components. The problemis prominent especially if the energy levels associated with the desiredsignal are low compared to the levels of coincident noise energy.

As an example we will consider the detection of relatively small,relatively faraway objects such as space debris with a radar. Thetransmitter of an ionospheric radar emits an electromagnetictransmission, usually a regularly repeated short pulse train, into ameasurement direction pointing to the sky. A radar receiver receivesechoes, which are the results of scattering of the transmission bymeteors, space debris, and other targets that are capable of interactingwith electromagnetic radiation at the frequency in use. Space debriscomes in sizes ranging from dust and paint flakes to complete bodies ofobsolete satellites. For the purposes of the present invention thesmaller end of the size scale is the most important, because of thelarge number (hundreds of thousands) and the difficult detectability ofsmall man-made objects orbiting the Earth. It is easy to understand thata radar echo produced by an object only some centimetres across at thedistance of several hundreds or even thousands of kilometres can not bevery powerful compared to measurement noise, even if very large (tens ofmetres in diameter) parabolic antennas are used.

FIG. 1 illustrates schematically an arrangement, in which a radarstation 101 has made measurements of the sky above. Each black dotrepresents an individual measurement. Depending on the characteristicsof the radar receiver, the signal processing capability, and thealgorithms available, each measurement may represent a combination ofdifferent measured quantities. Typical quantities to be obtained as rawdata are the round-trip delay it took for the transmission to betransmitted, scattered, and received; as well as the Doppler shift thatthe scattering target caused. From these the range (distance between theradar station and the target that caused the echo), radial velocity, andradial acceleration of the target can be calculated. The term “radial”refers to the direction of the straight line combining the radar and thescattering target. Radars equipped with monopulse feeds, as well asphased array systems, are also capable of measuring the angle or arrivalfrom a point target.

We assume that during the time interval under examination, exactly onesolid object orbiting the Earth has crossed the antenna beam. Some ofthe detected echoes were actually caused by said solid object, while theothers are false echoes that represent either actual scattering of theradar transmission but by non-orbiting objects (such as meteors), orsimply noise. The white dots marked with a vertical uncertainty bar arethe actual target-related measurements in FIG. 1, and the curve 102represents its orbit around the Earth. The problem is to decide, whichof the (potentially very large number of) measurements should actuallybe taken into account as representing the orbiting object. Each dot inFIG. 1 is drawn with a velocity vector that represents the velocity thatcan be read from the radar measurement for the corresponding echo. It isintuitively very easy to understand that the velocity vectors of theechoes related to the actual orbiting target follow quite closely itsorbit and are relatively close to each other in magnitude, while thevelocity vectors of the other echoes may have any arbitrary directionand magnitude.

Combining multiple measurements of a moving target into one unifieddescription of the target in terms of trajectory is a common problem inremote sensing. A wide variety of methods exists for solving thisproblem. Perhaps the most commonly used method is the so calleddetection threshold method, which relies on the fact that when a signalis strong enough compared to the noise level, it has to be a target witha very high probability. However, this approach suffers from severalshortcommings. It cannot cope very well with active radar jamming, andit cannot be used to detect weaker targets, as the false alarm ratewould be too large.

FIG. 2 illustrates schematically a similar problem that occurs incommunications. A transmitting device 201 uses original data 202 toproduce a transmission, which it emits in the form of a modulatedelectromagnetic carrier wave signal towards a receiving device 203. Inorder to find out the payload contents of the transmission, thereceiving device 203 produces a series 204 of measurements that reflectwhat was received. Each individual measurement may contain values of oneor more quantities such as phase, amplitude, and/or frequency. Again,only some of the measurements at the receiving device 203 are actuallyassociated with the original transmission, while others representinterference or noise. Again, for example if the transmitting devicewanted to conceal its transmission among noise to avoid detection byhostile parties, it may be difficult for the receiving device to decide,which measurements it should take into account for reconstructing theoriginal data.

SUMMARY OF THE INVENTION

An objective of the present invention is to present methods andarrangements for producing an organized subset from a multitude ofmeasurements, wherein each measurement is a value or a set of valuesthat describe characteristics of an assumed target, and wherein saidmultitude of measurements have been obtained by processing a receivedelectromagnetic signal.

An objective of the invention can also be described as to presentmethods and arrangements for producing a probable description for amultitude of measurements, wherein each measurement is a value or a setof values that describe characteristics of an assumed target, andwherein said multitude of measurements have been obtained by some formof remote sensing, e.g. by processing a received electromagnetic signalin a radar, sonar, or lidar system.

The objectives of the invention are achieved by arranging themeasurements in a ranked order, proceeding through the measurements insaid ranked order and each time testing, whether associating ameasurement with a particular target would increase the probability ofthe multitude of measurements describing the assumed behaviour oftargets.

According to one aspect of the invention there is provided a method forproducing an organized subset from a multitude of measurements, whereineach measurement is a value or a set of values that describecharacteristics of an assumed target, and wherein said multitude ofmeasurements has been obtained by processing a received electromagneticsignal, such as a signal received in one or more remote sensing systems,the method comprising:

-   -   arranging the multitude of measurements to a ranked order        according to a measurement-specific value, the magnitude of        which is assumed to correlate with a reliability of the        measurement,    -   initially designating individual measurements in said multitude        of measurements as not being associated with a target,    -   calculating an initial probability density    -   picking from said multitude of measurements a measurement that        is not associated with a target,    -   selecting from said multitude of measurements a candidate        correlating measurement,    -   calculating a probability density reflective of the picked        measurement and the candidate correlating measurement being        associated with a same target,    -   as a response to the calculated probability density being        indicative of higher probability than the initial probability        density, marking the picked measurement and the candidate        correlating measurement as being associated with the same        target, and    -   outputting, as the organized subset, those measurements that        have been marked as being associated with the same target.

According to another aspect of the invention there is provided anapparatus for producing an organized subset from a multitude ofmeasurements, wherein each measurement is a value or a set of valuesthat describe characteristics of an assumed target, and wherein saidmultitude of measurements has been obtained by processing a receivedelectromagnetic signal, the apparatus comprising:

-   -   a data arranging unit configured to arrange the multitude of        measurements to a ranked order according to a        measurement-specific value, the magnitude of which is assumed to        correlate with a reliability of the measurement,    -   a data designator configured to initially designate individual        measurements in said multitude of measurements as not being        associated with a target,    -   a probability density calculator configured to calculate an        initial probability density, and    -   a data selector configured to pick from said multitude of        measurements a measurement that is not associated with a target;        wherein:    -   said data selector is additionally configured to select from        said multitude of measurements a candidate correlating        measurement,    -   said probability density calculator is additionally configured        to calculate a probability density reflective of the picked        measurement and the candidate correlating measurement being        associated with a same target,    -   as a response to the calculated probability density being        indicative of higher probability than the initial probability        density, said data designator is configured to mark the picked        measurement and the candidate correlating measurement as being        associated with the same target, and    -   the apparatus is configured to output, as the organized subset,        those measurements that have been marked as being associated        with the same target.

According to yet another aspect of the invention there is provided acomputer program product comprising, on a computer-readable medium,machine-readable instructions that, when executed on a computer, causethe computer to implement a method for producing an organized subsetfrom a multitude of measurements, wherein each measurement is a value ora set of values that describe characteristics of an assumed target, andwherein said multitude of measurements has been obtained by processing areceived electromagnetic signal, the method comprising:

-   -   arranging the multitude of measurements to a ranked order        according to a measurement-specific value, the magnitude of        which is assumed to correlate with a reliability of the        measurement,    -   initially designating individual measurements in said multitude        of measurements as not being associated with a target,    -   calculating an initial probability density    -   picking from said multitude of measurements a measurement that        is not associated with a target,    -   selecting from said multitude of measurements a candidate        correlating measurement,    -   calculating a probability density reflective of the picked        measurement and the candidate correlating measurement being        associated with a same target,    -   as a response to the calculated probability density being        indicative of higher probability than the initial probability        density, marking the picked measurement and the candidate        correlating measurement as being associated with the same        target, and    -   outputting, as the organized subset, those measurements that        have been marked as being associated with the same target.

The method presented in this description differs from previous methodsas it utilizes a combination of a holistic Bayesian statistical targetmodel and a customized optimization algorithm that searches for the peakof the a posteriori probability density arising from the combined modeland prior distributions of the target trajectory parameters. In thisapproach, there is one single statistically and physically motivatedoptimality criterion, which determines when a target is detected.

The main challenge with the Bayesian probability density approach is thefact that the model space, i.e., the number of possible different modelsthat can explain the measurements, is too large to be exhaustivelysearched through. To address this problem, we have developed a customoptimization algorithm that only scans through likely regions of thesearch space, e.g., making use of the fact that moving targets that passthe radar beam are closely spaced together in time and space.

The exemplary embodiments of the invention presented in this patentapplication are not to be interpreted to pose limitations to theapplicability of the appended claims. The verb “to comprise” is used inthis patent application as an open limitation that does not exclude theexistence of also unrecited features. The features recited in dependingclaims are mutually freely combinable unless otherwise explicitlystated.

The novel features which are considered as characteristic of theinvention are set forth in particular in the appended claims. Theinvention itself, however, both as to its construction and its method ofoperation, together with additional objects and advantages thereof, willbe best understood from the following description of specificembodiments when read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates finding measurements that are associated with atrajectory of a target,

FIG. 2 illustrates finding measurements that are associated with atransmitted signal,

FIG. 3 illustrates a method and a computer program product according toan embodiment of the invention, and

FIG. 4 illustrates an apparatus according to an embodiment of theinvention.

DETAILED DESCRIPTION OF THE INVENTION AND EMBODIMENTS

An embodiment of the invention will be discussed in the framework ofspace debris investigations made with the EISCAT (European IncoherentScatter Radar) radar equipment.

Combining multiple detections of a moving target into one unifieddescription of the target in terms of trajectory is a fairly commonproblem in radar and telescopic measurements. A wide variety of methodsexists for solving this problem, but these are mostly optimized foron-line analysis of air traffic control radars with low false detectionrates.

We take a different approach to the problem by inspecting the globalprobability density of all measurements using the Bayesian framework,avoiding many heuristic processing steps and simplifying the problem. Wealso give one possible algorithm that can be used to search for themaxima of the probability density—or in this case, the set of targetsand their trajectories.

The current space debris analysis used for EISCAT measurements is athree-step procedure that involves:

-   1. Coherent integration in blocks-   2. Combining integration blocks into events-   3. Improving target detection accuracy.

Initially, a certain duration of time is coherently integrated. Inaddition to searching for the most probable Doppler shift and rangegate, different accelerations are also searched. If necessary, thecomputations in this stage can be accelerated with an algorithm known asFastGMF and described in the patent publication U.S. Pat. No. 7,576,688,which is incorporated herein by reference. Alternatively, the gridsearch can also be accelerated using the non-uniform (in time andfrequency) fast Fourier transform, as has been shown by Keiner, J.,Kunis, S., and Potts, D. in their study “Using NFFT 3—a software libraryfor various nonequispaced fast Fourier transforms ACM Trans. Math.Software”. The first step of the analysis procedure is reasonably welldescribed already in earlier studies but the second step has not yetbeen properly addressed until now.

The second step consists of determining which coherent integrationblocks (or measurements) belong to the same target and which integrationblocks do not contain anything meaningful. In this description, we focuson a model and an optimization algorithm that can be used for addressingproblems involved step two, in a close to optimal but computationallyefficient way.

Step three involves using the trajectory obtained in the detection step(2) to improve the target trajectory estimate. The optimal way would beto use the original raw voltage data, and fit a trajectory directly intothe raw voltage data. This can be done for example using MCMC (MarkovChain Monte Carlo) methods or using a combination of grid and gradientsearch methods. The computations can be sped up significantly by usingresults of the detection step as an initial guess for the targetparameters. We will not discuss step three in this description.

Moving Point Target Model

Our data is a set of N noisy measurements M=(m₁, . . . , m_(N)) oftarget trajectory related information, where each measurement cancontain several measurable quantities. Typically, the measurementm_(i)=(r_(i), {dot over (r)}_(i), σ_(i)) contains signal amplitudeσ_(i), range r_(i) and velocity {dot over (r)}_(i) at time instantt_(i). We will assume this is the case in the following description,although many of the derivations are also applicable in cases where noDoppler or amplitude values are available. The majority of thesemeasurements are not expected to contain information about any target atall, and they are expected to consist of instrumental noise, or moregenerally just something else than the desired signal. This set-up issimilar to what was described earlier with reference to FIG. 1 and FIG.2.

It is essential to note that the concept of “measurement” refers in thisdescription to a value or a set of values that describe characteristicsof an assumed target. Thus for example the momentary voltage value atthe output of a radar receiver is not a “measurement” in the sense ofthis description, but “raw data” or “raw voltage data”. As a comparisonto communications, where the “target” to be detected is the correctcontent of a transmitted symbol, a “measurement” would be an estimate ofsaid content. We assume that a multitude of measurements has beenobtained by processing a received electromagnetic signal

In the example above the measurement is a set of three values, namelyrange, velocity, and amplitude, of which the latter is a directindicator of the interaction cross section of the target with the radarsignal. It is typical to both remote sensing and communicationsapplications that in order to obtain a “measurement” a not insignificantamount of signal processing has to be performed already. Above suchsignal processing has been referred to as coherent integration inblocks, which is the case especially in remote sensing applications.

The problem is to determine the number N_(ev) of unique targets withunique trajectories that are contained in the set of measurements, theirapproximate trajectories, and which measurements contain informationabout each detected target. Bayesian model selection provides a way ofassigning a probabilty density for different models and their parameters

$\begin{matrix}{{{p\left( {k,\left. \theta^{(k)} \middle| D \right.} \right)} = \frac{{p\left( {\left. D \middle| k \right.,\theta^{(k)}} \right)}{p\left( \theta^{(k)} \middle| k \right)}{p(k)}}{p(D)}},} & (1)\end{matrix}$where p(k, θ^((k))|D) is the a posteriori probability density fordifferent models k and their corresponding model parameters θ^((k)).p(D|k, θ^((k))) is the likelihood function, which describes theprobability of the measurement given a model and set of modelparameters. p(θ^((k))|k)p(k) is the a priori density for the models andmodel parameters. p(D) is the probability of the data, and it can bethought of as a normalization constant.

In the case of detecting point targets, the number of different models kis astronomical. Assuming that each target n is described by a set ofunique measurements I_(n), with each measurement belonging to not morethan one target, the number of different models is given by the Bellnumber:

$\begin{matrix}{{B_{N} = {\sum\limits_{k = 0}^{\infty}\frac{k^{N}}{k!}}},} & (2)\end{matrix}$which grows extremely fast, e.g. B₂₀≈5.1·10¹³ and B₃₀≈8.5·10²³. Thismeans that in practice we cannot perform an exhaustive search throughall the possible models unless the number of measurements is very small.For larger sets of measurements, it is only possible to consider asmaller subset of all the possible models. This is why we have todevelop a algorithm that only goes through a small subset of the modelspace.

In order to evaluate p(D|k, θ^((k))), we have to be able to establish aforward theory that describes the measurements in terms of modelparameters. In this case, we assume that the target trajectory of eachdetected target can be described with some parametric function, e.g., inthe case of monostatic space debris measurements, the measured range andradial Doppler velocity can be described using a polynomial descriptionof the radial trajectory as r_(n)(t; θ_(n))=r_(n)+v_(n)t+a_(n)t² and{dot over (r)}_(n)(t; θ_(n))=v_(n)+a_(n)t, with target specificparameters θ_(n)=(r_(n), v_(n), a_(n)) and θ^((k))=U_(n)θ_(n), wherer_(n)(t; θ_(n)) signifies a description of the radial location, r_(n) isrange, v_(n) is scalar radial velocity, a_(n) is scalar radialacceleration, t is time, and {dot over (r)}_(n)(t; θ_(n)) signifies adescription of the radial velocity. The index k indicates one particulararrangement of the sets I_(n).

The particular polynomial description given above is just one choice toparametrize the trajectory, and it is not a limitation of the invention.In other situations the parametrization might take a different form inorder to describe the target better. For example space debris circlesthe Earth, which in a short timespan is well approximated by a steadyKeplerian orbit. In the case of a multi-static observation, it wouldadvantageous to use this parametrization for the trajectory. In areconnaissance radar different parametrizations could be employeddepending on whether the assumed target is an aeroplane, a sea-goingvessel, or a land vehicle. In a communications application it is oftenpossible to utilize typical regularities in the transmitted signal topresent a parametrization of its assumed behaviour as detected by thereceiver, e.g. by employing a Markov Chain communication model. Theparametrization does not need to be linear, but choosing a linearparametrization if one is available may simplify the calculationsconsiderably.

Coming back to the space debris measurements as an example, and assumingthat our measurements consist of range r_(i) and Doppler velocitymeasurements {dot over (r)}^(i), we can now express the forward theoryas

$\begin{matrix}{r_{i} = \left\{ {\begin{matrix}{{{r\left( {t_{i};\theta_{n}} \right)} + \xi_{i}},} & {i \in I_{n}} \\{v_{i},} & {i \notin {U_{n}I_{n}}}\end{matrix},{and}} \right.} & (3) \\{{\overset{.}{r}}_{i} = \left\{ {\begin{matrix}{{{\overset{.}{r}\left( {t_{i};\theta_{n}} \right)} + \xi_{i}^{\prime}},} & {i \in I_{n}} \\{v_{i}^{\prime},} & {i \notin {U_{n}I_{n}}}\end{matrix},} \right.} & (4)\end{matrix}$where ξ_(i) and ξ′_(i) are the measurement errors for the range andradial velocity measurements. Measurements that don't belong to anytarget detection are described with the random variables v_(i) andv′_(i), which have a distribution that models the instrumental noise.For example, in the case of space debris measurements, this is veryclose to uniformly distributed noise.

Assuming that the range and velocity errors are zero mean and Gaussianwith ξ_(i)˜N(0,

_(i) ²) and ξ_(i)˜N(0,

_(i) ²), we can write the likelihood of our measurements asp(D|k,θ ^((k)))=κCexp{−S},  (5)where the sum of squares term is

$\begin{matrix}{S = {{\sum\limits_{n = 1}^{N_{ev}}{\sum\limits_{i \in I_{n}}{\frac{1}{2\vartheta_{i}^{2}}\left( {r_{i} - {r_{n}\left( {t_{i};\theta_{n}} \right)}} \right)^{2}}}} + {\frac{1}{2\varrho_{i}^{2}}\left( {{\overset{.}{r}}_{i} - {{\overset{.}{r}}_{n}\left( {t_{i};\theta_{n}} \right)}} \right)^{2}}}} & (6)\end{matrix}$and C is the normalization factor

$\begin{matrix}{C = {\prod\limits_{n = 1}^{N_{ev}}\;{\prod\limits_{i \in I_{n}}\;\frac{1}{2{\pi\vartheta}_{i\;\varrho\; i}}}}} & (7)\end{matrix}$

and κ contains the probability of the non-targets, i.e. the probabilitydensity of v_(i) and v′_(i). In this case, we assume that they areuniformly distributed,κ=(Δ_(r)Δ_(v))^(−N) ^(n) ,  (8)with prior ranges Δ_(r)=r_(max)−r_(min) and Δ_(v)=v_(max)−v_(min). Thenumber of measurements that contain actual events is

$N_{p} = {\sum\limits_{n = 1}^{N_{ev}}\;{\# I_{n}}}$and the number of non-events events is N_(n)=N−N_(p).

The prior distribution for our model space p(k)=B_(N) ⁻¹ is assumed tobe uniform, giving each model equal probability. Also, the priordistribution for model parameters p(θ^((k))|k) is assumed to beuniformly distributed:p(θ^((k)) |k)=(Δ_(r)Δ_(v)Δ_(a))^(−N) ^(ev) ,  (9)with the additional uniform distribution for the acceleration parameterΔ_(a)=a_(max)−a_(min).

The logarithmic posteriori density, leaving out the constant terms, cannow be written aslog p(θ^((k)) ,k|D)=−S−αN _(n) −βN _(ev)+log C,  (10)whereα=log Δ_(r)Δ_(v)  (11)andβ=log Δ_(r)Δ_(v)Δ_(a)  (12)

The complete result would be to study the full posterior probabilitydistribution, but because of the vast search space, this would bedifficult or impossible to do in most practical cases. Instead, wesuggest searching for the peak of the distribution:({circumflex over (k)},{circumflex over (θ)} ^((k)))=arg max_(k,θ)_((k)) p(θ^((k)) ,k|D).  (13)

Doing even this exhaustively may require a discouragingly large amountof resources, as there is a very large number of models—going throughe.g. the more than 10²⁷⁴ possibilities required for 200 measurements isnot possible in practice at least at the time of writing thisdescription. However, it is usually not necessary to exhaustively searchthrough all models in order to come up with meaningful results. In thecase of radar measurements of space debris and meteor head echos, theevents are localized in time and range, and additionally we also have anestimate of the errors related with each measurement as they depend onsignal power. Using this information, it is possible to sort the modelspace in terms of relevance, so that the most probable areas withtargets are processed first and the areas that are the least likely tocontain meaningful targets are processed last, and to obtain areasonably good estimate of equation (13).

The specific formulation of the a posteoriori density in equation (10)results in a simple logarithmic probability density form that can beefficiently evaluated. The formulation starts by assuming thatmeasurements either belong to a moving point target, or they don't. Themoving targets are modeled using some parametric trajectory, while themeasurements not containing any targets are assumed to span a finiteparameter space uniformly.

The a posteriori probability density does not necessarily need to resultin a form of Eq. (10). In some cases it can be more profitable to modelthe trajectory using another parametric form, or to model thenon-targets using a different distribution than the uniform distributionthat resulted in Eq. (10). E.g., in the case of active radar jamming,the distribution of Doppler and range measurements can be completelydifferent than in the case no jamming, where only ground clutter andreceiver noise are the main contributing factors.

Restricted Grid Search

In order to maximize the logarithmic probability of the measurements, wehave devised a heuristic search algorithm that utilizes the fact thatmeasurements with large signal power give better estimates, and the factthat targets passing the radar are localized in range and time.

The algorithm initially sorts all measurements in decreasing order ofmeasured signal power σ_(i). We start with the assumption that allmeasurements are non-targets, and calculate the initial posterioriprobability density. Then we go through each measurement in decreasingorder of signal power, and attempt to fit the trajectory model tomeasurements that are temporally close by. These attempts are made oneby one, solving the maximum likelihood parameters for S with matrixequations at each step. The measurements are marked as belongingtogether if the posteriori probability density is increased. Ifmeasurements are marked as belonging to some event, they will not beused for other events.

FIG. 3 illustrates a method and a computer program product for therestricted grid search. At step 301 the measurements are arranged to aranked order according to a measurement-specific value, the magnitude ofwhich is assumed to correlate with a reliability of the measurement. Inthe case of locating orbiting point targets with radar, themeasurement-specific signal power is a natural candidate. In some othercases the value used to decide the order could be selected differently.For example a radar, sonar, or lidar arrangement could be used to detectan approaching missile or other threatening object, the typical velocityof which can be known with certain reliability. In such a case thehighest-ranking measurements could be those where the velocity valuecontained in the measurement was closest to the assumed actual velocity.

Steps 302 and 303 are initializing steps where all measurements areinitially designated as not belonging to or being associated with anytarget (step 302) and the initial probability density is calculated(step 303). The last-mentioned gives a kind of a threshold value thatcan be used for comparisons, because it indicates, how probable itshould be to obtain the current set of measurements by just looking atthe clear sky with no orbiting objects currently in view. In order toaccept a later found combination of two or more measurements as anindication of a point target that actually crossed the radar beam, aprobability value higher than at least the initial probabilitycalculated at step 303 should be obtained.

The currently highest-ranking measurement that has not yet been markedas belonging to (or associated with) a particular target is picked atstep 304, and a candidate correlating measurement is selected at step305. In this description we use consistently the verb “to pick” and itsderivatives to refer to the measurement picked at step 304, and the verb“to select” and its derivatives to refer to the candidate measurementselected at step 305. Once a particular measurement has been picked asthe picked measurement, a number of candidate measurements can beselected at their turn in order to find out, which of them (if any) canbe associated with the same target as the currently picked measurement.

The selection of the candidate correlating measurement is made accordingto a criterion that again should involve some insight about how thesought-after targets actually behave. In radar measurements of spacedebris it is natural to assume that if a particular target produced aclear echo at some time t, it will also produce a similar echo at aslightly differing time t±Δt. Again thinking about the approachingmissile as an alternative example, the candidate correlating measurementcould be selected on the basis that it was made in the same or onlyslightly different spatial direction. The invention does not limit thecriterion that is used to select the candidate correlating measurementat step 305, as long as it reflects the above-mentioned insight of howactual targets should behave.

More generally we may consider a particular picked measurement assetting up a “neighbourhood” within a coordinate system defined bymeasurement-specific characteristics. As an example, a measurement maycomprise measurement-specific values for time (the exact time at whichthe measurement was made), distance (the distance at which themeasurement indicates a target to be), and velocity (the velocity atwhich the measurement indicates a target to move). A neighbourhood setup by that measurement in the time-distance-velocity coordinate systemwould include such other measurements that consist of sufficientlysimilar time, distance, and velocity values, taken the assumed behaviourof the target. In other words, only such other measurements are includedin said neighbourhood that could possibly relate to the same target. Ifan approaching missile is looked for, and a picked measurement indicatesan approaching velocity of 600 m/s at a distance of 15 kilometres, it isnot reasonable to think that another measurement indicating awithdrawing velocity of 100 m/s at 20 kilometres only shortly thereafterwould relate to the same target. Therefore said other measurement wouldobviously not be included in the neighbourhood of the currently pickedmeasurement.

In mathematics it is customary to use the concept of norm toinvestigate, how close to each other two points are in some coordinatesystem. In any coordinate system, there exist multiple ways of defininga norm, A very frequently used norm is a the Euclidean norm, which isthe square root of the sum of squares of the coordinates. Thecalculation of an Euclidean norm may be weighted, if it is assumed thatthe coordinates involved come with differences in significance. Forexample, if a measuring arrangement is assumed to produce more accuratevalues of distance than velocity of targets, it may be advantageous togive the distance coordinate more weight in defining the neighbourhoodof a measurement and calculating how close two measurements are to eachother in said neighbourhood. Numerous other ways of defining andcalculating a norm are known and can be used.

Since a norm can express the “closeness” of two measurements veryconveniently even with a single value, a predetermined limit may be setto an acceptable neighbourhood of a measurement by giving a limitingvalue for the norm. A measurement is then closer than said predeterminedlimit to another measurement, if the value of a predetermined normcalculated for these two measurements exceeds a limiting value thatcorresponds to said predetermined limit. The definition of the norm tobe used, as well as the limiting value for the norm, reflect knowledgeof the assumed behaviour of the target. Therefore by changing thedefinition of the norm, and/or by changing the limiting value, the samemethod can be applied to different kinds of applications.

The definition of the norm, and its known association to the assumedphysical characteristics of the target, may involve different limitingconsiderations for the different coordinates on which the norm is based.These limiting considerations may even vary dynamically according to thevalues included in the currently picked measurement. As an example, ifthe currently picked measurement includes a velocity value 1 km/s (i.e.if the currently picked measurement is assumed to represent a targetthat is moving at 1 km/s), it is clear that within a time frame of, say,±5 seconds, the target could not have moved very much further than 5 kmin distance. Some other picked measurement might include a velocityvalue 2 km/s, so plausible candidate measurements to be associated withthe same target might well have a distance coordinate differing by 10 kmwithin the same time frame of ±5 seconds. At step 306 the maximumlikelihood parameters are solved for S, typically with matrix equations.The calculation of the current probability density is shown separatelyas step 307. In figurative terms, the probability density calculated atstep 307 is reflective of the picked measurement and the candidatecorrelating measurement being associated with a same target and tells,how probable it would be to obtain the current set of measurements, ifduring the time when the measurements were obtained, there was a targetin view which had the range, velocity and acceleration values asindicated by the currently picked measurement and the candidatecorrelating measurement. Therefore the calculated probability density iscompared to the highest previously calculated probability density atstep 308. If the calculated probability density represents an increaseto the highest previously calculated probability density, themeasurement picked at step 304 above and the candidate correlatingmeasurement selected at 305 above are marked at step 309 as belonging toa common target. In other words they are not any more “free”measurements, where “free” means not associated with any target.

Speaking of “higher probability” is customary, but it should not beunderstood restictively, because in some calculating algorithms thesigns and/or the value axes have been selected so that actually asmaller value indicates higher probability. Therefore we may generalizeby saying that the calculated probability density may be indicative ofhigher probability than the previously calculated probability density.

Possibly looping back from step 310 to step 305 means that once ahighest measurement was picked at step 304, other free measurements(e.g. all remaining free measurements that are closer than apredetermined limit to the picked measurement, in the sense of apredetermined norm) are gone through in order to find all thosemeasurements, the grouping together of which increases the probabilitydensity. When this run-through has been made for a particularmeasurement picked above at step 304, a transition to step 311 occurs,where it is checked, whether free measurements were left that did notbelong to any of the targets found so far. If yes, a return to step 304occurs where the highest-ranking measurement still free is picked.Thereafter the inner loop consisting of steps 305 to 310 is againrepeatedly circulated in order to find matching pairs for the currentlypicked measurement from among the remaining free measurements in itsneighbourhood.

The concept of a neighbourhood is very practical in limiting the amountof calculations that need to be made. In principle it would be possibleto go through all possible combinations of all measurements with eachother, in order to find out the one for which the probability densitywould be the highest. In practice this is possible only for very smallsets of measurements, as has been shown above with reference to formula(2). Selecting the candidate measurements only from the neighbourhood ofthe picked measurement represents an important part of the concept ofrestricted grid search.

If there are sufficient resources (time and processing power) forcalculations, it is advantageous to define a relatively largeneighbourhood. This ensures that as many as possible of the measurementsthat actually relate to a common target will be found. After all,various error sources affect the eventual values that will be containedin the measurements, which means that measurements that quite certainlyrepresent the same target may occur even relatively far from each otherin the sense of the norm.

According to an embodiment of the invention, observing the closeness ofmeasurements with each other and setting a limit to how many candidatemeasurements are gone through can be separated from each other. As anexample, the norm may have been defined as an Euclidean norm that takesinto account time, velocity, and distance. Candidate measurements aregone through in an increasing order of their norm value, i.e. byproceeding within the neighbourhood of the picked measurement from theclosest possible candidate measurement outwards. However, the limit ofhow many candidate measurements will be gone through is not defined as amaximum value of the whole norm, but simply as a maximum value of time.Other measurements will simply not be selected as candidate measurementsif they have been made more than, say. ±5 seconds away from the pickedmeasurement. This example illustrates the choice of using threedifferent coordinates to calculate the norm but only one of them to setthe limiting value.

The limiting value may be as simple as a counted number of candidatemeasurements, which counted number may or may not have some dynamicrelation to other characteristics of the set of measurement data. As anexample, one may terminate looping back from step 310 to step 305 when0.1% of those measurements have been selected as candidate measurementsthat are closer in time to the picked measurement than half a minute.

The algorithm continues to process all or some significant portion ofthe measurements and will at some point terminate, as illustrated by thearrow down from step 311. If the computation is halted at some point oftime, e.g., due to real-time processing requirements, the algorithm hasalready processed the echos with the smallest estimations errors, i.e.,the ones most likely to contain a target. As a result, there is outputan organized subset that contains those measurements that have beenmarked as being associated with the same target. In case several targetswere found, there are output the measurements themselves as well as anindication of which measurements were associated with a common target ineach case.

FIG. 4 is a schematic illustration of an apparatus according to anembodiment of the invention. The apparatus is configured to receive datathrough a data input 401, which is for example a wired or wirelessconnection to a computer, or an interface for receiving portable datastorage means, a local area network connection or a wide area networkconnection. Especially the apparatus is configured to receive as data alarge number of measurements, so that each measurement has been obtainedby processing a received electromagnetic signal, and each measurement isa value or a set of values that describe characteristics of an assumedtarget. At least a large majority of the measurements should eachcontain a measurement-specific value, the magnitude of which is assumedto correlate with a reliability of that particular measurement. Theapparatus comprises a measurement data storage 402 configured to storethe received measurements.

The apparatus comprises a data arranging unit 403, which is configuredto arrange the measurements to a ranked order according to saidmeasurement-specific value, the magnitude of which is assumed tocorrelate with a reliability of the measurement. The apparatus comprisesalso a data designator 404, which is configured to initially designateindividual stored measurements as not being associated with a target.Further parts of the apparatus are a probability density calculator 405,which is configured to calculate an initial probability density, and adata selector 406, which is configured to pick from said multitude ofmeasurements a measurement that is not associated with a target andadditionally configured to select from said multitude of measurements acandidate correlating measurement. The probability density calculator405 is configured to calculate a probability density reflective of thepicked measurement and the candidate correlating measurement beingassociated with a same target. The data designator 406 is configured to,as a response to the calculated probability density being indicative ofhigher probability than the initial probability density, mark the pickedmeasurement and the candidate correlating measurement as beingassociated with the same target.

The concepts of picking and marking of measurements may be conceptuallyimaged as utilizing a memory 407 of organized data sets, in which one ormore data sets are stored. In this conceptional thinking, a data set inthe memory 407 consists of measurements that have been marked as beingassociated with a common target. In a way, the data arranging unit 403,the data designator 404, and the data selector 406 cooperate to gothrough the initially anonymous multitude of measurements in themeasurement data storage 402, so that as a result, those of themeasurements that have true meaning eventually end up intowell-specified entities within the memory 407 of organized data sets.Physically this does not need to mean transferring any measurementsbetween actual memory circuits or locations, because memory managementtechniques known from prior art allow organizing and handling recordsstored in a memory as logical entities that logically belong to acertain part of the memory or logically change into a different part ofthe memory, even if physically they stay stored at the one and the samelocation of physical memory.

The apparatus is configured to output, as an organized subset, thosemeasurements that have been marked as being associated with the sametarget. In the block diagram the apparatus comprises a data output 408dedicated to this purpose. The output 408 may be for example a wired orwireless connection to a computer, or an interface for writing data intoportable data storage means, a local area network connection or a widearea network connection. Outputting data may comprise displaying some ofthe data or its derivatives on a display device.

Blocks 403, 404, 405, and 406 are advantageously implemented asmachine-readable instructions stored on a memory medium, so thatexecuting said machine-readable instructions by a processor causes theapparatus to implement those steps that have been described earlier in amore detailed manner as a method. Blocks 402 and 407 are advantageouslyimplemented as machine-readable and -writable memory means together withtheir associated memory management software and hardware.

Further Considerations

In a basic form the restricted grid search selects candidatemeasurements beginning from the one which is closest (in the sense ofthe applied norm) to the currently picked measurement. If theprobability density calculated after selecting the closest candidatemeasurement is higher than the initial probability density, the closestcandidate measurement is marked as being associated with the same targetas the currently picked measurement. After that the next closestmeasurement is selected as the candidate measurement.

A false association with the same target is possible. We may consider acase where measurement X was picked, and measurements A and B are foundin its neighbourhood, A being the closest. In reality, measurements Xand B come from the same target, but measurement A does not.Accidentally it happens, however, that the probability densitycalculated first for the association of measurements X and A is higherthan the initial probability density. Consequently measurement A becomeserroneously associated with the same target as measurement X. Then, whenmeasurement B is selected as the candidate measurement and the nextprobability density is calculated for all measurements X, A, and B, alower value is obtained (because in reality, A and B were not related atall). What happens is that measurement B is rejected, and the erroneousassociation X+A is maintained.

These kind of errors can be avoided by calculating, after each selectionof a candidate measurement, the probability density for all possiblecombinations of the selected candidate measurement, the candidatemeasurements previously marked as associated with the same target, andthe picked measurement. However, the number of calculations becomes veryeasily prohibitively large, unless an equally large amount of processingpower is available. One way for checking for errors might be to do,among the group of candidate measurements marked as associated with thesame target, a number of random modifications and to look, whether anyof them further improves the calculated probability density.

The present invention may take advantage of the calculation algorithmpreviously known from the U.S. Pat. No. 7,576,688. In its original form,said algorithm gives out an individual measurement, which in the senseof statistical analysis is the one within a number of measurements thathas the highest probability of representing an actual target. For thepurposes of the present invention, any number of measurements indecreasing order of probability can be taken out of said algorithm, forlater use as picked measurements. By using the original measurementdata, a neighbourhood can be set up around each measurement soidentified, in order to find out, whether there are more measurementsthat could be associated with the same target as the picked measurementin question.

If the present invention is applied to telecommunications, one mayconsider reconstructing a transmitted message at the receiver using aso-called Markov chain approach. Most original signals involve some kindof regularity, like the known frequent occurrence of certain characterstogether. A part of the received signal that represents a character (ora short character string) can be thought of as a “measurement” in theparlance used earlier in this description. If one such part has beenreceived and decoded, associating a further measurement with it could beconsidered as increasing the probability density if, taken the knownregularity laid down by some known basic property of the message, afurther message part derived from that particular measurement couldoccur together with the first part at a high probability.

As an example, we may consider that said known basic property of themessage is its language, say, French. A part of the signal has beenreceived and decoded, and found out to contain the character string“qu”. In French it is fairly common that the next character followingthat particular string is a vowel, mostly “e” or “i”. Thus if receivingand decoding a further part of the signal gives such a vowel, it can beassociated with the first part with relatively high certainty. In otherwords, associating the character string “qu” and for example furtherpart “e” with the same “target”, i.e. the same portion of the receivedsignal, probably results in decoding this portion of the received signalcorrectly.

In the foregoing description we have frequently referred to using thereceived, measurement-specific power level as the criterion of arrangingthe set of measurements into ranked order. In doing so it should betaken into account that at least in some cases there may be strongsignals present that are not related to the desired targets at all. Forexample in wireless telecommunication applications in hostileenvironments there may be jammer signals, which are powerful radiosignals produced by an adverse party and aimed at disruptingcommunications. In such cases it may be advisable to use some othermeasurement-specific value than power as the criterion for arranging themeasurements into ranked order. If power is still used as the criterion,the known presence of high-power unwanted signals underlines thesignificance of making the restricted grid search cover also the lowerend of the ranked order as far as computationally possible, because itmay happen that measurements actually describing the desired targetsonly appear a way down the ranked order.

A specific area of application of the present invention is constitutedby multi-instrument measurements. In many cases a number of differentinstruments (and/or a number of separate channels used by a singleinstrument) measure the same targets. A multi-instrument measurement maybe for example a combination of two radars located at differentlocations, operating at possibly different frequencies. It is possibleto combine measurements from these two different systems within oneanalysis using the same algorithm. In multi-instrument measurements careshould be taken in defining the norm that is used to set up theneighbourhood of a measurement, so that factors that can be consideredmost reliable have the highest significance in the norm. For example ifthe instruments are separate radars, the distance coordinate istypically more reliable than the time and velocity coordinate, so onemight consider weighing especially distance in the definition of thenorm. A natural measurement ranking measure would in this case be thesum of the two signal powers. One particular advantage of the presentinvention it its ability to produce organize subsets from a large set ofmeasurements, where even a very large majority of the originalmeasurements may actually come from no target at all. The organizedsubsets produced according to the invention contain, with highprobability, a large portion of those measurements that actually do comefrom a target. The invention ensures that the production of suchorganized subsets can be accomplished with a reasonable demand ofprocessing power and calculating time.

What is claimed is:
 1. A method for producing an organized subset from amultitude of measurements, wherein each measurement is a value or a setof values that describe characteristics of an assumed target, andwherein said multitude of measurements has been obtained by processing areceived electromagnetic signal, the method comprising: arranging themultitude of measurements to a ranked order according to ameasurement-specific value, the magnitude of which is assumed tocorrelate with a reliability of the measurement, initially designatingindividual measurements in said multitude of measurements as not beingassociated with a target, calculating an initial posterior probabilitydensity, picking from said multitude of measurements a measurement thatis not associated with a target, selecting from said multitude ofmeasurements a candidate correlating measurement, calculating aposterior probability density reflective of the picked measurement andthe candidate correlating measurement being associated with a sametarget, as a response to the calculated posterior probability densitybeing indicative of higher probability than the initial posteriorprobability density, marking the picked measurement and the candidatecorrelating measurement as being associated with the same target, andoutputting, as the organized subset, those measurements that have beenmarked as being associated with the same target.
 2. A method accordingto claim 1, wherein: at the step of picking a measurement, thehighest-ranking measurement in said ranked order that is still notassociated with any target is picked.
 3. A method according to claim 1,wherein: as a response to the calculated posterior probability densitybeing indicative of lower probability than the initial posteriorprobability density, the current candidate correlating measurement isreplaced with another selected candidate correlating measurement, andthe steps of calculating a posterior probability density reflective ofthe picked measurement and the current candidate correlating measurementbeing associated with a same target, as a response to the calculatedposterior probability density being indicative of higher probabilitythan the initial posterior probability density, marking the pickedmeasurement and the candidate correlating measurement as beingassociated with the same target and as a response to the calculatedposterior probability density being indicative of lower probability thanthe initial posterior probability density, the current candidatecorrelating measurement is replaced with another selected candidatecorrelating measurement are repeated, while maintaining the same pickedmeasurement, until essentially all measurements that were not yetassociated with any target and that are closer than a predeterminedlimit to said picked measurement have been selected at their turn as acandidate correlating measurement, wherein a measurement is closer thansaid predetermined limit to another measurement if the value of apredetermined norm calculated for these two measurements exceeds alimiting value that corresponds to said predetermined limit.
 4. A methodaccording to claim 3, wherein: at the step of replacing the currentcandidate correlating measurement with another selected candidatecorrelating measurement, said another selected candidate correlatingmeasurement is a measurement that is still not associated with anytarget and that has not yet been selected as a candidate correlatingmeasurement for the currently picked measurement, and the selection ofthe candidate correlating measurement is made according to an assumedbehaviour of the sought-after targets.
 5. A method according to claim 3,wherein: after essentially all measurements that were not yet associatedwith any target have been selected at their turn as a candidatecorrelating measurement while maintaining the same picked measurement,said picked measurement is replaced with the highest-ranking measurementin said ranked order that is still not associated with any target, thecycle of picking every time the highest-ranking measurement in saidranked order that is still not associated with any target and repeatingthe steps of claim 3 with the picked measurement is repeated until apredetermined ending criterion is fulfilled, and as the organizedsubset, there are output all those measurements that were marked asbeing associated with the same target as some other measurement,together with an indication of which measurements were associated with acommon target.
 6. A method according to claim 4, wherein said endingcriterion is fulfilled at the occurrence of at least one of: allremaining measurements still not associated with any target have beenselected at their turn as candidate correlating measurement for eachpicked measurement, or a time limit for producing said subset expires.7. A method according to claim 1, wherein calculating a posteriorprobability density means calculating the logarithmic posteriorprobability density log p(θ^((k)), k|D) aslog p(θ^((k)) ,k|D)=−S−αN _(n) −βN _(ev)+log C, where k is an index ofpossible models that explain the association of measurements withtargets, θ^((k)) signifies the model parameters of a k:th possiblemodel, D signifies the data, i.e. the multitude of measurements,$S = {{\sum\limits_{n = 1}^{N_{ev}}{\sum\limits_{i \in I_{n}}{\frac{1}{2\vartheta_{i}^{2}}\left( {r_{i} - {r_{n}\left( {t_{i};\theta_{n}} \right)}} \right)^{2}}}} + {\frac{1}{2\varrho_{i}^{2}}\left( {{\overset{.}{r}}_{i} - {{\overset{.}{r}}_{n}\left( {t_{i};\theta_{n}} \right)}} \right)^{2}}}$n is a summing index, N_(ev) is the number of unique targets that arecontained in the set of measurements according to a currently selectedmodel, i is a summing index, I_(n) signifies the multitude ofmeasurements,

_(i) ² is the variance of a first quantity, such as range, in ameasurement, r_(i) is the value of the first quantity, such as range, inan i:th measurement, r_(n)(t_(i); θ_(n)) is a parameterizedrepresentation of an assumed behaviour of the first quantity for then:th target

_(i) ² is the variance of a second quantity, such as velocity, in ameasurement, {dot over (r)}_(i) is the value of the second quantity,such as velocity, in an i:th measurement, {dot over (r)}_(n)(t_(i);θ_(n)) is a parameterized representation of an assumed behaviour of thesecond quantity for the n:th target α=log Δ_(r)Δ_(v) β=logΔ_(r)Δ_(v)Δ_(a) Δ_(r) is the assumed range of allowable values for thefirst quantity, Δ_(v) is the assumed range of allowable values for thesecond quantity, Δ_(a) is the assumed range of allowable values for athird quantity, such as acceleration, in a measurement, N_(n) is thenumber of non-target measurements that are contained in the set ofmeasurements according to a currently selected model, and C is anormalization factor.
 8. A method according to claim 1, wherein: themeasurements are value sets indicative of spatial location and dynamicmovement of targets measured with a radar, sonar, or lidar.
 9. A methodaccording to claim 1, wherein: the measurements are value setsindicative of at least one of frequency, amplitude, and phase of symbolstransmitted as sequences of electromagnetic oscillation.
 10. Anapparatus for producing an organized subset from a multitude ofmeasurements, wherein each measurement is a value or a set of valuesthat describe characteristics of an assumed target, and wherein saidmultitude of measurements has been obtained by processing a receivedelectromagnetic signal, the apparatus comprising: a data arranging unitconfigured to arrange the multitude of measurements to a ranked orderaccording to a measurement-specific value, the magnitude of which isassumed to correlate with a reliability of the measurement, a datadesignator configured to initially designate individual measurements insaid multitude of measurements as not being associated with a target, aposterior probability density calculator configured to calculate aninitial posterior probability density, and a data selector configured topick from said multitude of measurements a measurement that is notassociated with a target; wherein: said data selector is additionallyconfigured to select from said multitude of measurements a candidatecorrelating measurement, said posterior probability density calculatoris additionally configured to calculate a posterior probability densityreflective of the picked measurement and the candidate correlatingmeasurement being associated with a same target, as a response to thecalculated posterior probability density being indicative of higherprobability than the initial posterior probability density, said datadesignator is configured to mark the picked measurement and thecandidate correlating measurement as being associated with the sametarget, and the apparatus is configured to output, as the organizedsubset, those measurements that have been marked as being associatedwith the same target.
 11. An apparatus according to claim 10, whereinthe apparatus is a remote sensing apparatus configured to receiveelectromagnetic signals from a remote target and to process the receivedelectromagnetic signals to form said multitude of measurements.
 12. Anapparatus according to claim 10, wherein the apparatus is acommunications apparatus configured to receive electromagnetic signalsfrom a remote transmitting device and to process the receivedelectromagnetic signals to form said multitude of measurements.
 13. Acomputer program product comprising, on a non-transitorycomputer-readable medium, machine-readable instructions that, whenexecuted on a computer, cause the computer to implement a method forproducing an organized subset from a multitude of measurements, whereineach measurement is a value or a set of values that describecharacteristics of an assumed target, and wherein said multitude ofmeasurements has been obtained by processing a received electromagneticsignal, the method comprising: arranging the multitude of measurementsto a ranked order according to a measurement-specific value, themagnitude of which is assumed to correlate with a reliability of themeasurement, initially designating individual measurements in saidmultitude of measurements as not being associated with a target,calculating an initial posterior probability density picking from saidmultitude of measurements a measurement that is not associated with atarget, selecting from said multitude of measurements a candidatecorrelating measurement, calculating a posterior probability densityreflective of the picked measurement and the candidate correlatingmeasurement being associated with a same target, as a response to thecalculated posterior probability density being indicative of higherprobability than the initial posterior probability density, marking thepicked measurement and the candidate correlating measurement as beingassociated with the same target, and outputting, as the organizedsubset, those measurements that have been marked as being associatedwith the same target.